section \<open>Example 3.8\<close>
theory Ex2
imports "../LCF"
begin
axiomatization
P :: "'a \<Rightarrow> tr" and
F :: "'b \<Rightarrow> 'b" and
G :: "'a \<Rightarrow> 'a" and
H :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and
K :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b)"
where
F_strict: "F(UU) = UU" and
K: "K = (\<lambda>h x y. P(x) \<Rightarrow> y | F(h(G(x),y)))" and
H: "H = FIX(K)"
declare F_strict [simp] K [simp]
lemma example: "\<forall>x. F(H(x::'a,y::'b)) = H(x,F(y))"
apply (simplesubst H)
apply (induct "K:: ('a\<Rightarrow>'b\<Rightarrow>'b) \<Rightarrow> ('a\<Rightarrow>'b\<Rightarrow>'b)")
apply simp
apply (simp split: COND_cases_iff)
done
end